Sydney recovered from conceding a penalty just six minutes in, when Vedran Janjetovic saved the spot kick, before Carney gave his side the lead by heading in from a well-worked move.
The Chinese side still had chances, but could only finish off one of them, with Tardelli making up for an earlier miss to seal a 1-1 draw for the hosts.
Shandong had struggled domestically so far, but the Chinese Super League outfit started brightly and had the visitors' defence under early pressure.
The Sky Blues were caught out on seven minutes when Janjetovic came well off his line and caught Wu Xinghan in the box.
But Janjetovic redeemed himself by saving Walter Montillo's penalty.
The away side recovered and just before the quarter hour, Sydney struck with a calculated move on the edge of the box.
Milos Dimitrijevic played in Riley Woodcock who crossed in for Carney to head home at the far post and make it 1-0.
Shandong had further scoring chances before the break, but couldn't trouble Janjetovic, despite a good opening for star striker Tardelli.
After the interval, Sydney had a header from a corner cleared off the line, while the home side went agonisingly close when Zheng Zheng's long-range bullet struck the crossbar and bounced on the line and out.
The home side finally the made the pressure count on 56 minutes. The Sky Blues defence failed to clear a corner and Tardelli latched on to a flick-on to level the score at 1-1.